OTANIEMI FINLAND
Report No. 8
On Ship Manoeuvring and Waterway Width
JUHANI SUKSELAINEN
Thesis for the degree of Doctor of Technology to be presented with due permission
for public examination and criticism in the Auditorium Ko 216 at the Helsinki
University of Technology on the 26th of May, 1975, at 12 o'clock noon.
Page 15 4th row
or
instead of
of
Page 23 3th row
-0.970
instead o-F
-0.0970
Page 25 eq. 3.14 should read
X =
(LX
Ô +X"
u2
+LX"
vr
X"y2
ç)h L uç
uuç
vrç
vvçç
Page 48 8th row
-For
instead
o-Ffo
Page 49 to 55 figures 6 to 12
115
000
tdw
instead
o-F115 tdw
Page 77 19th row
Y"instead of
Y"The present study was initialized in 1971 with the main emphasis on the experimental sector. It was soon discovered, that the central problem o-F waterway width and alignments had to be sought elsewhere. Thus the originally intended experimental program was reduced and more interest was devoted to theoretical considerations and computer
simulations and analysis. The work has been carried out in the Ship Hydrodynamics Laboratory of The Helsinki University o-F Technology during the years 1971 to 1974. The numerical calculations have been performed on a UNIVAC 1108 system of The Ministry of Education.
I wish to express my gratitude to Professor V. Kostilainen, principal of the laboratory, for his warm and encouraging support during every phase of my work.
I also thank the staff of the Ship Laboratories of The Helsinki University of Technology for their valuable help. Especially the advice and assistance of Messrs. P. Hervale, i. Holmström, 0. Huuska, and H. Lindroos is warmly appreciated. I also want to express my appreciation for the valuable help of Mr. T. Ovaske in the experimental phase of the work.
My thanks are also to Mr. H. Benson for correcting the. English text of the manuscript.
The financial support of the Finnish Board of Navigation has made the present study possible, and is gratefully acknowledged.
CONTENTS
PREFACE 3
SYMBOLS 7
i . INTRODUCTION 9
REVIEW OF EXISTING WATERWAY DESIGN GUIDELINES AND
RESEARCH 10
2.1 Waterway dimensioning methods 10
2.2 Manoeuvring simulator experiments in waterway research 12
A PROBABILISTIC APPROACH TO WATERWAY DIMENSIONING 15
3.1 General aspects 15
3.2 Equations o-F ship motion 15
3.2.1 Newtonian equations of motion 15
3.2.2 Treatment of hydrodynamic Forces 18
3.2.3 Determination of hydrodynamic coefficients 21
3.3 Confined water effects 24
3.4 Outer disturbances 26
3.4.1 Wind forces 26
3.4.2 Flow effects 28
3.4.3 Wave induced motions 29
3.5 Modelling the ship steering 32
3.5.1 The human controller 32
3.52
The closed-loop control system 333.5.3 Ship positioning 35
3.6 Monte Carlo techniques 36
3.6.1 General principles 36
3.6.2 Distribution selection and parameter estimation 37
3.5.3 Ship control philosophy 42
3.7 Computational scheme 45
3.7.1 Choosing the simulation system 45
3.7.2 Simulation details 45
4. SIMULATION RESULTS 49
4.1 Straight course manoeuvring space 49
4.2 Manoeuvring lane in bends 51
4.3 The influence of speed and water depth 53
4.4 Conclusions 56
SUMMARY 57
REFERENCES
APPENDIX I Determination of the coefficients in the ship
equations of motion by system identification SS
APPENDIX II Simulation model input coefficients 77
A area
B ship beam
C coefficient
block coefficient coefficient vector
c mean flow velocity at rudder
F fetch F Froude number f frequency control vector g gravity constant h water depth I moment of inertia
J propeller advance coefficient
K control gain
k radius of gyration, propeller coefficient
L ship length
m ship mass
N yaw moment
n propeller revolutions per second
P Norrbin's P-factor p propeller pitch O torque R turning radius Rn Reynolds number r yaw rate
S spectral density, propeller slip ratio
s, y* curvilinear guide line coordinates
T draught, thrust, period
t time, thrust deduction factor
U ship velocity
u component of U in x-direction
speed loss in x-direction
y component of U in y-direction
W channel or ship lane width
w wake fraction
x.y,z Orthogonal coordinates of a right-handed system oF body axes x0,y0,z0 Orthogonal coordinates o-F a right-handed system of space axes
X" etc. nondimensional hydrodynamic coefficients
Y force in y-direction
Y system output vector
y5 bank distance
half-width of dead space in observation of y*
a deflection angle o-F waterway bend
B drift angle
rudder angle
water depth parameter T/(h - T)
n bank clearance parameter = B/ye
A wave length
p engine setting, direction angle
p mass density
a standard deviation
ship heading angle
heading relative to guide line
V volume displacement SUBSCRIPTS A wind C current d derivative E engine e equilibrium F friction h hydrodynamic L lateral o observed P propeller p proportional Q torque R relative sg steering gear T transverse, thrust W wave
1. INTRODUCTION
Ship manoeuvring research has traditionally been directed towards deterministic aspects of individual vessel behaviour. The results achieved have only had a very limited influence on waterway plann-ing methods as applied by civil engineers. The present state of art in ship manoeuvring could be much better utilized to achieve a rational fairway design approach, however. Perhaps the lack of cooperation among the various disciplines directly involved: naval architects, nautical experts, and civil engineers, has been a reason. As the creation of economical waterways is a relatively important national goal for a country like Finland, all available knowledge should be concentrated to attain a favourable solution for each project.
One of the most important observations in recent years concerning various aspects of ship control, has been the recognition of the
human nature of the ship operator. The control behaviour of a helmsman and / or pilot is still very poorly understood as well as the human perception of navigational markings and aids. New research facilities established in recent yeard can certainly contribute to better knowledge, but in the present situation it is most important at least to recognize these aspects as relevant in connection with Fairway design.
The present study has as its goal a rational method for waterway dimensioning using the available knowledge on steered ship manoeuvring behaviour. When a more precise description of the human ship controller is available, the method could be adjusted to take this into account. In other respects many simplifying assumptions have also been unavoidable in the manoeuvring model, but these can undoubtedly be reconsidered as new information emerges. The important question of waterway depth and depth
margins will not be covered by this study. Neither are the special problems of two-way traffic with interaction effects btween ships included.
2. REVIEW OF EXISTING WATERWAY DESIGN GUIDELINES AND RESEARCH
2.1 Waterway dimensioning methods
In this connection only the most common type of Finnish coastal waterway is considered. It is called the open-type channel and is shown in Fig. 1. The main feature of this kind of fairway is ti-e lack of visible banks and thus suitable navigational aids are required for position determination. This type of waterway is also open to cross currents and wave action, which are not encountered in a restricted-type channel which has continuous visible banks on each side.
SHIP LANE
SHIP CLEARANCE
CHANNEL
WIDTH W
Fig. 1 Open-type channel.
one-way traffic the total channel width is usually divided into a manoeuvring or ship lane with a bank clearance lane on each side /1/. In two-way traffic two ship lanes and an additional ship clearance lane are required, Fig. 1. The width of the manoeuvring
lane is assumed to be dependent on the controllability of the vessel concerned and values of 1.4 to 2.0 times ship beam are suggested. The required bank clearance depends on such circumstances as cross winds and currents, ship speed, visibility, material of
channel banks, etc. The proposed values are from O.6B to 1.5B. The width of 3D m is proposed for the ship clearance lane. The
BANK CLEARANCE
dimensions presented here are based on investigations carried out studying the sea level Panama canal project /2/. The P.I.A.N.E. criteria /3/ give fairly similar channel width values, 3.. .4B for
one-way traffic and 6.. .76 in two-way traffic in ideal conditions. These criteria have also been applied to the design of the Finnish coastal and inland waterways. Using the methods published by P.I.A.N.0. there have been soma efforts to study the width of the manoeuvring lane by estimating separately the required width for
the uncontrolled yawing of the design vessel as well as the additional space needed for the steady drift angle /4/.
The current design guidelines of the Finnish Board of Navigation
The two last items try to allow for considerable inaccuracies in navigation and refer to natural fairways.
The need of widening a manoeuvring channel at curves is universally recognized, but no adequate standard is available for the actual
lay-out. Some formulas proposed date as far back as 1926 /1/, but they must be considered rather unsatisfactory. A physically reasonable formulation for the Pequirad increase of channel width in bends has been proposed by Kray /6/, and can be written:
AW 2000 c U2 L2
gRO S
S = undisturbed sight distance from the bridge of the ship.
C = coefficient of the controllability of the ship (poor = 1;
good = 2; very good = 3). This formula is far more rational than the aforementioned, which only contain the parameters L and R. It must be recognized, however, that the parameter C makes the application rather difficult, should one always have the poorly manoeuvred ship in mind? Also its background is not explained any
further in /6/, but obviously it is mainly based on work carried
(2.1)
/5/ for the fairway width are as follows:
minimum W = lOT 4B
narrows W = lOT 4B
normal W = 30T 2 12B
out in connection with the sea level Panama-canal project /2/. As mentioned in the reference /6/, the model experiments with
manually steered models in channel bends gave widely distributed results for the required width. Therefore caution is required in implementing the Eq. (2.1).
Another unsolved question in designing channel bends is the
distribution of the extra width. In Finland the method of allowing two-thirds at the outer edge end only one-third at the inner edge is generally accepted /7/. To assist actual dredging work the theoretically curved channel is usually approximated by polygonal banks.
The minimum radius of curvature in bends shculd normally be at least 1OL, which value is required by the Finnish Board of
Naviation /5/ and is also widely accepted /1/. Un inland waterways a smaller radius is often tolerated and on Lake Saimaa a value of EL has been used /7/. The angle of deflection of the curve is also very important from the manoeuvring point of view. The current Finnish practice tends to limit the deflection angle to 30° /7/ and this value has also received acceptance as the upper limit among the vast majority of ship navigators /8/. At certain locations such a limitation of the angle is not applicable, but the manoeuvrability of such places should then be secured with additional precautions to ensure the safe passage of ships.
2.2 Manoeuvring simulator experiments in waterway research
In recent years a number of ship manoeuvrlirg simulators have been established /9, 10, 11/, mainly intended for training navigators. The appearance of VLUC- and more recently ULUC-vessels especially has necessitated adequate schooling to the very slow time scale as characteristic to their steering. A ship manoeuvring simulator consists of a wheelhouse with its controls and instruments, a projection system for generating an image of the ships bow with the appropriate marine scenery visible through the bridge windows, and the heart of the system, a computer, which does the actual simulation and communicates to the operator using the requisite mentioned.
Besides teaching marine personnel, a great deal of experimentation has been made with these simulators concerning the behaviour of a helmsman, as well as uf the manually steered ship. This kind of work has been carried out especially in Holland by several investigators: Hooft /12/, Wagenaar /13/, and Oldenkamp /14/ to mention a few. Also in Cothenburg, Sweden, Norrbin /15/ and van Berlekoni /15/ have studied these phenomena by using a simulator.
As regards waterway dimensioning, the papers
of
Hooft /17/ and Oldenkamp /14/ are the most interesting. They introduce astatistical concept to the determination of the required width
of
a manoeuvring lane. Only straight line motion is considered. however. They conclude that a ship's transverse position can be regarded as normally distributed and using the distributionof
maxima, they determine the required ship lane width in terms o-Flocal standard deviation
of
the ship's lateral position in a series o-F simulations.The use
of
manual steering simulators has certain drawbacks in this kind of study, however. No test run can be repeated, the test person's skill increases with number o-F runs, difficulties ofgetting an adequate number of able test persons., the long time required for performing the run in real time, and finally the high cost of running such a system. Of course very valuable data can be acquired concerning the manual control problem and human behaviour. This knowledge is again essential for developing an accurate simulation model to be executed using a computer, and using an automatic controller instead of a human one. Unfortunately the data available comparing the performance o-F manual ship steering and autopilot steering is very scarce. Truijens et al. /16/ and van Berlekom /16/ have treated the problem in studying the step response behaviour of a steered ship. The last mentioned results must be used with caution, however. The reason is that van Berlekom
used an unoptimized autopilot, which is only mentioned in the original report /19/.
The key factor in making comparisons between ship controllers is obviously the number o-F merit applied. There are of course several
leading line, steering gear loading, steering induced resistance, and helmsman fatigue to mention the most important ones. In this study only the two first mentioned are considered. To properly balance all the requirements is a very delicate problem and
General aspects
All the present day fairway dimensioning rules are based on real operating experience of definite model experiments. The
deterministic model experiments certainly give some qualitative results for the given waterway, but as found in the aforementioned rnanoeuvring simulator studies, the representation of the human control system of the real ship is unsatisfacotry in the contracted model time scale /18/. Dimensioning methods which try to sum up the worst case values of different disturbing effects like wind, waves and current caused drift or uncontrolled yawing caused by the properties of the closed loop control system incorporating the human controller with his perceptual qualities. The former ones have been estimated in many cases fairly accurately and
theoretically well explained, but the uncontrolled yawing has only been treated with uncertain empirical formulae, the applicability of which to other ship types is at least uncertain. The summation of the worst case disturbances leads in many cases to excessive fairway width requirements /4/, which must be reduced by uncertain reasoning to achieve an agreement with established dimensioning guidelines. This kind of work is in many cases normal engineering
practice and often leads to sound results. It does not explain the real structure of the very phenomena, however, and especially the
stochastic nature of ship passages through a certain portion of waterway remains completely obscure.
The Dutch simulator studies /14/ have brought the random nature of ships' paths at a certain fairway station under the magnifying glass. With the high expenses of real time simulation, only very moderate series of runs have been made. Computer simulations offer much more liberal treatment of large samples of runs with randomly variable parameters like weather, ship manoeuvring qualities,
navigation errors, tidal currents etc. Such a model could be capable of using a much larger sample with statistically moe significant results. It could be used as well for waterway dimensioning, as for assessment of required manosuvring qualities
for a certain waterway leg. A computer simulation study certainly must also have data from simulator runs with manual control to test its ultimate validity as such a design tool, but even as a crude test version it could perhaps shed light on certain hydro-dynamically predestinated statistical qualities of ships in certain well defined manoeuvring situations. Also tha importance of the accuracy of positioning systems could be tested with such a simulation model. So it was decidsd to build a suitable model and test its capabilities and to compare the results with existing waterway design guidelines and formulae. The general structure and background of various parts of the model is presented in this chapter and the results of computations are presented in chapter 4.
A simulation model is always a compromise between the requirements of accuracy and reasonable computer time Especially when using Monte-Carlo modelling, the amount of computer time required grows to considerable amounts with the complexity of the model used. Fortunately no strict limitations were set in respect to the available computer time. Thus it was possible to pay most attention to the accuracy of the model. Special attention was to be paid to the determination of the hydrodynamic derivates using mainly free sailing model data.
3.2 Equations for ship motion
3.2.1 Newtonian equations of motion
In the field of ship hydrodynamics the following axis system is generally agreed upon /20/: Origin at the center of gravity of the ship, x-axis directed towards the bow, y-axis to starboard, and z-axis downward. An additional coordinate system is nominally fixed in relation to the earth, the positive z0-axis is vertically downward and the x0-axis lies in the general direction of initial motion (Fig. 2).
In studying the basic steered motion of a surface vessel the roll, pitch, and heave motions are generally assumed negligible /21/. Thus the Newtonian equations o-P motion of the ship can be written
74
Fig. 2 The coordinate system.
in the body-fixed coordinate system:
m(u-rv) (3.1)
mlv ur) =
+ Y (3.2)
(3.3)
If the ship is moving in deBp open water the position and heading do not influence the forces. The hydrodynamic forces on the right side are usually assumed to be functions of the instantaneous values of the variables in brackets. Experimental research
suggests that these forces are non-linear functions of velocities and the control variables, but are linearly dependent on the accelerations /22/.
3.2.2 Treatment of hydrodynamic forces
Abkowitz /21/ introduced the method of expressing Xh.
h' and Nh
as Taylor expansions around the point (u = U0, y = r = = 0). Derivatives up to third order are considered. Because of the symmetry of a normal ship, the odd derivatives in the X-equation and the even derivatives in Y- and N-equations can be discarded. Neither nonlinear acceleration derivatives nor interaction between acceleration and velocity derivatives can be expected if a
potential theory solution is taken as basis /21/.
In the Abkowitz-model the non-dimensionalizing of the coefficients or derivatives is usually performed by the following units: unit of length = L, unit of time = L/U, and unit of mass = .5 p L3. This in turn means that forces are non-dimensionalized with division by
22
.5 p L U and when U diminishes towards zero the method fails.
Therefore the non-dimensionalizing method proposed by Norrbin /15/ is more natural. Here the units are: unit of length = L, unit of time = (L/g)5, and unit of mass = m, ship's mass. In this system the forces are simply related to the ship's weight. All
coefficients in the equations of ship motion in this study are non-dimensionalized by using this method.
Norrbin /15/ also prOposes the use of quadratic terms instead of cubic ones present in the Abkowitz-model. This method is well supported by the general dependence on pressure forces on the square of the flow velocity. Mathematically the use of second power terms is of course a bit less convenient because of the need
of absolute value functions for taking care of the appropriate
sign.
As stated before Abkowitz developed the Taylor expansion around the point u = U0. If it is now intended to cover the whole speed range including zero speed, it is apparently more convenient to use the point u = O as the neutral point. In this case the
equations can be written by using u instead of u.
It must be pointed out, however, that the use of Taylor-expansions in forniulati.ng the hydrodynamic forces and moments for the ship manoeuvring equations has no physical meaning. It is merely a method for representing a function of several variables in close proximity to the chosen neutral point. As stated before and recognized by Kettenis /23/ for practical manoeuvring simulations one needs a sufficiently accurate representation of the hydro-dynamic forces in a broad speed range. So if the function of
several variables is to be expressed using the instantenous values of state variables, the most natural way of doing this is the use of polynomials incorporating these variablee. Thus the derivatives are substituted by polynomial coefficients /24/. Some investigators
/25/ try to subdivide the hydrodynamic forces into three groups: hull, propeller, and rudder effects. This is rather misleading, however, as the hull coefficients include the contributions of one or more propellers and undeflected rudders /26/. Thus such grouping should be avoided as interaction is unavoidably present also in terms where the actual element is not explicitly represented.
The thoughts outlined above lead to a formulation of ship equations of motions, which is essentially similar to the one developed by Norrbin /15/ and used by Berlekom /27/. The hydrodynamic forces and moment in Eqs. (3.1 to 3.3) are written:
m Xh - (LX. ù + X" u2 + (Lg)1 X" u4 + Lg X (1 - t) + L X" vr u uu uuuu vr X"vv y2 + X«
dci
O2 + X cicißo) = X o ici 00dci
6 hOy = (LY + LY" ur + Y" uy * Y"
vivi + Y" ciclo
h L y ur uy ivi y cic 0
+ Y" c ici ii lOi gL Y X ) = h0 cicigji loi Nh m (L' N. * L N ur + N" uy + L N" ivi r N" r ur uy vir ciclo * N" c
cici
ißi loi
1016161 101 +gLNX) = NhQ
The non-dimensional propeller thrust force can b written using the polynomial:
X (gL) (T" u2 + L T" un + L2 T" ini n)
uu un min
The propeller revolutions are determined by the torque balance of the propulsion system. The torque absorbed by the screw can be represented by the polynomial:
-1 « 2 « 2
= (gLi (Q u * L Q un + L
'1
n in! n)
uu un
For a diesel engine the net torque given can be written:
1/2 -3/2
= p (L g Q n * Q
+ F sign(n)
n p
The net torque o-F the propeller-engine system gives the differential equation for the number o-F revolutions:
= g L1 (Q" + Q) (2iî (3.10)
Finally an equation is written to approximate the mean effective
(3.4)
(3.5)
(3.6)
(3.8)
rudder in-Flow velocity squared: 2 «
2.
2 c L[ un + L C n i-F n O un nfl 2 c=0
The propeller thrust and torque equations, in case of a fixed-pitch propeller, can easily be deduced from the k-J diagram for
the screw /15/. If a [P-propeller is used, the pitch change ue to the control system can as well be inforporated in the approximative equation, or the pitch setting can be used as an additional input parameter.
Norrbin /15/ has derived the rudder in-Flow velocity in Eq. (3.11) based on the simple actuator disc theory, which gives the
slipstream velocity:
a k 1/2
c = (1 -w) u (1
+
-The wake -Fraction outside the screw race is assumed to be 4/3 times the mean value. The mean square inflow velocity is then calculated -For the rudder. Okada /24, 25/ has suggested a simpler empirical method, for single screw ships:
= (1-w)
u (1 6.2553/2)1/2 (3. 13)3.2.3 Determination of hydrodynamic coefficients
The coefficients in the three equations of ship motion (Eqs. (3.1
to 3.6)) are at the present state of art to be deduced from model experiments. The important questions of scale effects and model-ship correlation in manoeuvring experiments shall not be discussed
here, as they are widely handled in several papers in Ref. /20/. In this respect it is only possible to try to get the most
dependable material available. The limitations of the methods must be accepted as well as their virtues if better ones are not
available.
if n < O
(3.11)
The manoeuvring model tests /29, 30/ can be divided into two main categories: captive model tests and free sailing model tests. In the captive model technique the ship model motion is the input and the hydrodynamic reactions on the hull, the output. The most
simple type of these tests is oblique towing. The required
requisite consists of a normal towing carriage equipped with three-component dynamometer to measure Xh.
h' and Nh. The model is
towed with varied drift and rudder angles and a suitable propeller RPS. Oblique towing tests can deliver all those coefficients in the equations of ship motion,.which do not relate to accelerations or angular velocity r.
The rotating arm facility is intended for steady state determin-ation of rotative coefficients. Here the model makes steady turning motion with varied R, 9, and S. The three plane reactions are
measured with a balance. These tests deliver ll steady state coefficients in the equations of motion, the rotative ones directly and the y- or 9-coefficients as interpolated between port and starboard turning when plotted as a function of the resiprocal
of R.
Planar motion mechanisms (PMM) or X-Y--towing carriages finally make the determination of all ship model coefficients possible. This is attained by use of unsteady, usually harmonic model motions in the measurements. The X-Y-carnage method in a wide tank can also act as an rotating arm facility and both types are naturally suited to oblique towing tests.
The free sailing model tests are conducted in a maneuvering basin or lake using mostly remote controlled ship models. Here only the model movements can be measured. Direct modelling of definite ship maneuvers is easily realized, but the determination o-P hydrodynamic coefficients is more complicated and is discussed in Appendix I.
The analytical tools available can only provid.e the coefficients
Y and N and the corresponding cross coupling coefficients Y and N , which are required in Eqs. (3.5 and 3.6) in case of
pronounced force and aft unsymmetry of the hull /15/. The two-dimensional zero frequency potential flow solution, integrated
over the ship length, gave the following comparison for the Mariner-ship (see Appendix I).
Y computed value -0.0970 , measured value /22/ = -0.938
V
= -0.0545 , n = -0.0414
r
The theory of ship manouvering does not provide many tools for modifying the coefficients in the basic equations of ship motion to improve the model-ship correlation. Actually only an
approximative adjustment of rudder coefficients is possible. Propositions have been made for improving the model-ship correlation by introducing special vanes to modify the flow at model stern /31/. It must be recognized, however, that there is
neither theoretical knowledge nor sufficient empirical data for a serious application of this method.
An important point in scaling results of free sailing model maneuvring experiments is the propulsion machinery employed. Its characteristics determine two major qualities: rudder inflow velocity and speed loss in a manoeuvre. As the former in no case can be accurately modelled due to scale effect on propeller load-ing and wake, the only possibility is to analytically take care of the inevitable differences between model and ship. The latter point could of course be modelled using an electronic programmable power unit to control the model propulsion motor to simulate the appropriate Q-n characteristics of the full-scale ship. As this would be rather complicated and still solve only part of the problem even in connection with a separate friction correction force applied on the model through an air propeller etc., it is more convenient to try to separate both effects as proposed by Norrbin /15/. This means the introduction of the propeller inflow velocity c instead of u in the rudder terms of the equations o-F motion as well as the use of the three auxiliary euations for
thrust, revolutions, and the inflow velocity c.
For qussi-steady conditions with a fixed machinery setting it would be of course possible to simplify the three propulsion dependent equations to only one. This would mean an expression for
c in terms of forward velocity u alone according to lines used by Strm-Tejsen for the approximative formulation of the thrust in the X-equation /22/. For transient manoeuvres including engine setting or propeller pitch variations as well as the use of occasional propulsion bursts to augment the rudder effect, the complete group of equations is required.
Some preliminary model tests were made to study the dependence of the propeller inflow velocity c on propeller loading. These
indicated that in model scale, the empirical formulation of Okada /28/ gave the best fit, but the actuator disc method failed in the form presented in Ref. /15/. A somewhat better agreement with experimental data was attained by use of an attenuation factor for the slip stream. This was also applied by Mori and Tanaka /32/ in
their ship manoeuvring simulation program.
For the purposes of the present simulation model the following sources for the various hydrodynamic coefficients in the equations of ship motion were chosen. Oblique towing was the only type of captive model tests feasible with the facilities of the Ship Hydrodynmics Laboratory, and so it could be used for the linear motion coefficients. The acceleration coefficients had to be
determined theoretically. The remaining coefficients had to be determined using free sailing model techniques. The method applied
is discussed more in detail in Appendix I. The coefficients in the auxiliary Eqs. (3.7 to 3.11) can, as stated before, be computed using ship propulsion data.
3.3 Confined water effects
The general nature of Finnish coastal and inland fairways is shallowness combined with occasional width limitations or dredged open-type channel sections. So the most important feature in a
ship motion simulation model is the inclusion of a dependence on
water depth. The hydrodynamic influence of local shoalings or natural banks is in most cases negligible /19/, but in certain specific cases it must as well be modelled.
The confinement parameters defined by Norrbin /15, 33/ are
regarded physically well founded and they are applied also in this study. The definitions are:
depth parameter ç T / (h - T)
bank distance parameter n = B /
The influence of the water depth parameter ç on the hydrodynamic forces and moment in Eqs. (3.1 to 3.3) can be written /27/:
22
m
Xh - - ç(L X. u + X" u2 + L X" + X" y ç
ç L uu vr
Y = ç(LY'-' + LY" ur' + Y" uy + Y" viv
hç L vç urç uvç
vvç
+ Y" C IC B BI Ici B IBI ç Nhç m ç (L2 N' ' + L N" ur + N" uy + L N"Ivi
r rç urç uvç vi rÇ + N" c Ici B161 ii
cIcIBIBI
i6IçLocal bank effects can be represented in the open-type channel at the present state of art only as an external force and moment /15, 33/, which depend on both confinement parameters:
m'
2 'T'hçn = L 'T'uu (ç,) u 2 Nhçn = m N" (ç,p) u (3. 14) (3.15) (3.16) (3.17) (3.18)The experimental data by Schoenherr /34/ and the theoretical work of Newman /35/ supply the basic data in the case o-if a vertical sidewall. Gawn /36/ gives experimental results for the cese of a sloping bottom. In channel type sections the data published by Eda /24/ can be regarded as useful,
3.4 Outer disturbances
The outer disturbances affecting ship manoeuvres are wind, current and waves. The order of importance is often the same, at least in the Finnish coastal waters, where the current velocities are relatively modest and no oceanic wave action is present. In other
circumstances the order of importance can be different. Flow conditions can be especially very difficult at certain geographic locations. In the following, the effects of these three natural powers and their implementation in the simulation model are
separately discussed.
3.4.1 Wind forces
So far the wind influence on ships has only bean studied from a deterministic point of view. Eda /37/ studied ship controllability
in seeking limiting wind velocity - ship speed relations for straight course motion. Similar lines have been followed by Ogawa /38/ who computed the influence of wind on turning circles. InDue and Ishibashi /39/ have also concentrated on definite manouvres.
In the present study the limiting values are useful only in the respect that they clearly show their unimportance in normal fairway traffic. The maximum wind velocity recorded at Katajaluoto off Helsinki during the period 1959-72 in a single case was 23 m/s out of some 25 000 measurements (see section 3.6.2). As the normal ship speed on coastal fairways is B to 16 knots the speed ratio does not exceed 6. Thus the most important Feature of wind influence is its general disturbing effect on the behaviour of a steered ship. From this point of view the generally gusty nature of the wind should also be modelled instead of the simple mean value, which has been considered by the above mentioned
authors.
Wind force and moment on ships have been studied by many investigators. Isherwood has recently summarized these in his
x)
The ferries with a large windage can have problems, however, if required to proceed with drastically reduced speed.
4 K x
SA(f) - 2 5/6
f (2 + X where
x = 1 219.
K = surface drag coefficient
= mean wind speed at 9.1 m
paper /40/. Only wind tunnel experiments made in homogenious air flow are considered by him. Some authors have tried to reproduce the atmospheric boundary layer in their experiments. As all used different velocity profiles it would be rather difficult to compare them with any accuracy. On the other hand the apparent wind as meeting a vessel running at normal to half speed (16 to 8 kn) is
always the vector sum of the true wind velocity and the ship speed caused wind, Fig. 2. This means a speed dependence on the
apparent wind pro-File. Thus one is certainly on the safe side if a uniform wind profile is stipulated. The paper by Isherwood /40/ gives the X- and Y-force as well as the N-moment coefficients in
the form of multiple regression polynomials. The heeling of a ship caused by wind influence can be neglected in the case of normal merchant vessels as it has been observed /41/ that the influence
of a small list on ship manoeuvring is negligible.
There is neither experimental nor theoretical data on th effect of gusty wind on ships. In the field of civil engineering a considerable effort has been directed to study the instationary effect of wind on tall buildings and other structures. Vellozzi Oohen /42/ have studied the structure of the wind field frani an engineering point of view. They use the gust spectrum by Harris as basic distribution:
(3.19)
A value of 0.010 has been used for K, which has been given for rough wooded country, as it can be assumed that it corresponds to the circumstances in the coastal waterways between islands.
In considering the effect of wind gustiness, the correlation between velocity fluctuations of a given frequency located at two separate points in space must be taken into account. As the
of a ship are small compared to its length, and the correlation. in vertical direction much better than in cross-wind direction, the correlation needs to be computed only in the horizontal plane. Considering the gustiness in an area with a diameter equal to the ship's length, the correlation function can be expressed /42/:
-2y -2p R(f)
(21-1+2
)(2U 212 where y = 11.5 f L/li4 p= 3.85 fL!4
(3.20)The gust spectrum is now weighted with R(f) and for practical computation divided into five discrete frequencies, which in turn are added t the mean wind velocity. As steady flow values are used for force and moment coefficients the method is a quasi-steady one, but as a correlatedgust spectrum only contains low frequencies it can be regarded as justified.
The force and moment coefficients in the simulation model are expressed as Fourier series of the fifth order of the apparent wind direction. The polynomials computed by Isherwood /40/ have
supplied the basic data for the coefficients.
3.4.2 Flow effects
Only homogenous current is considered in this study. This is due to lack of adequateknowledge on inhomogeneous flows and their influence on hydrodynamic forces affecting ship motions. Some methods of the unsteady wing theory could perhaps be tried, but as
the zero-aspect ratio wing analogy works rather badly even in steady state /15/. it does not seem very promising. On the other hand the real flows in rather shallow coastal waters also have a strong vertical velocity gradient, which makes the task still more complicated. Thus it seems that such conditions only can be
accurately studied with models. The steady flow model, however, can give useful qualitative results in studying local currents when used pulse-vice at a certain portion of the fairway. In that
case one is always on the conservative side in predictions.
In homogeneous current the ship velocity vector is computad in a coordinate system moving with the water. Then the vector of the
current speed is added to it. In other respects the steady current has no effect on the ship equations of motion /15/.
L3
Wave induced motionsIn ship hydrodynamics the motions in a seaway are computed using differential equations of motion with frequency dependent
coefficients for inertial, damping and corresponding cross-coupling terms. In ship manoeuvring theory, however, constant coefficients are applied, but a nonlinear treatment is required in most cases, whereas in seakeeping studies, a linear approach is usually
adequate except for severe rolling motion. One possibility to cope with the problem of combining the two motion mode models would be a modification of the wave excitation terms to give correct sway and yaw motions with zero-frequency coefficients in the equations of motion. Eda & Crane /43/ tried to simplify the problem still more by using only rough approximations for sway and yaw
excitations without any corrections. Reasonable agreement with experimental data was reported.
As the sway and yaw motions are strongly coupled with the rolling of the ship all these three modes should be considered in an appropriate model. The roll motion, however, is usually not included in the manoeuvring equations of motion. So it appears that a suitable method of treatment for wave induced hip motions is a separate computation model for oscillatory sway, yaw, and roll. The sway and yaw motions are then superposed on the low frequency motions. The existence of published computing algorithms for ship motions in a seaway makes this approach still more
attractive.
The seakeeping calculations for the purpose of this study were made using the program package "Scores' published by the Ship Structure Committee /44/. The hydrodynamic coefficients are computed using
the Franks close-fit method /45/. This gives considorably better accuracy than the simple Lewis-form approach as used in the
original "Scores package. Especially at the end sections, the Lewis-form fit is poor. It must be considered, however, that
close-fitting requires, by an order of magnitude, more computer time. For a normal ship hull about 30 minutes is required by the UNIVAC 1O8.
The actual implementation of the computing method includes the approximation of a one-dimensional sea spectrum with five discrete wave lengths in the region À/L = .4 . .. 1.5 . A table o-f sway and
yaw responses as well as the phase difference is computed for the five wave lengths and for every ten degrees of wave direction angle. The simulation program uses this table as input data and interpolates linearly in the table. Originally, Fourier as well as polynomial fitting was tried to the data, but without complete success. The tabular form is of course more economical from the computing time point of view, but necessitates more input data for
the simulation program.
For simulation of wave influences in coastal waterways it is necessary to include a fetch dependency in the wave spectrum used. As the fetches in our coastal waters are always rather small, it is stipulated that the fetch alone determines the sea state for a given wind speed and the swell is ignored. This is mainly because of lacking data on actual wave-wind correlation in Finnish waters. This method certainly exaggerates the waves at early hours of higher winds, but as again the remaining swell is ignored, these effects are mutually compensated. Additionally, if the wave damp-ing effect of smaller rocky areas, shallows, and small islands which at least to some extent protect the coastal fairways, is ignored, the approach of computing the wave induced ship motions in the lateral direction icon the conservative side. Due to a lack of experimental data no corrections are applied to the wave spectra due to the shallovrness of the water.
A convenient form of wave spectrum with limited fetch has been proposed by Silvester /46/. The constants in the Pierson-Moskowitz-spectrum
(2r)
here = wind velocity at 19.5 m height = 1.08 times the velocity at 10 m height.)
are expressed -For a limited fetch as -Follows:
-0. 194
= 0.0081 (F / FFAS)
8 = 0.1 exp (2.00 (F / FFAS)°284)
The fetch -For fully arisen sea FFAS is given in nautical miles by the expression
FFAS = 8.54 (3.24)
The equation (3.21) is only a rough approximation, but -For the present purpose it is adequate, as the effect of waves on larger
vessels is ultimately rather small and unimportant. When more accurate data is available, the model can be easily updated.
The wave induced mean drift force and moment are not included in the simulation model. This is partly due to a lack of adequate data for vessels with non-zero speed and partly due to the limited
importance of these factors. If considered necessary a percentage increase o-f wind effects can be substituted for the wave drift effects /4/.
The added resistance in waves has also been neglected in the present model as its computation would be rather complicated and its importance in the coastal waters of the Baltic Sea is small and on the other hand the accuracy of modelling the X-equation is
of lesser importance than that of the Y- and NI-equatiOns, which more directly determine the width of the lane required by the ship.
(3.22)
3.5 Modelling the ship steering
3.5.1 The human controller
Certainly the most difficult task in formulating a simulation model for' a manoeuvring ship, is the description fo the behaviour of the human controller, which in most cases consists of two persons. A mate or pilot makes the decisions and determines the desired path of the ship. The helmsman acts according to given orders and additionally takes care of keeping a desired course or line. The use of a separate helmsman is an inheritance from old sailing days before any power assisted steering. At that time the whole effort of one or even several hands were required to handle the helm, but nowadays it is merely a habit to have a helmsman. His duties ould to a still larger extent be executed by an
automate and in more difficult manoeuvring situations handling of all the bridge functions could be done by two officers or a pilot and a mate. This would require a central control stand, where all primary control equipment is placed in a convenient form around the operator /47/.
When considering the present day ships, the pilot / helmsman combination is the controlling element, the control behaviour of which should be modelled to reach the ultimate goal of the present
study. The role of a helmsman alone has been studied fairly intensively during the last ten years. Both in zapan /48/ and in the Netherlands /49/ models have been derived for the behaviour of the helmsmen when steering a ship along a straight course. Th emergence of ship steering simulators in Sweden and Holland has had a a stimulating effect on these studies. Using a simulator it is possible to repeat an experiment in identical conditions. This means of comparing the performance of a test person in several
runs clearly demonstrated the difficulties involved. In the first few runs the learning behavior of the test person leads to de-creasing errors, but suddenly an unexpectedly bad run may follow. Also the experience of the test person has a considerable influence on the test result /18/.
nfluencing factors can be found /50/:
- purely hydrodynamic conditions - control conditions
-_ navigation conditions.
0f these three, the first one can be modelled with considerable accuracy, as nearly all the effort directed towards ship manoeuvr-ing has been concentrated on this theme. The control action of a human helmsman has so far been approached mainly on the basis o-E a
linear controller model originally developed for airplane and motor car steering /51/. It has been shown by Veldhuyzen Stassen /52/, however, that in manual VLCC steering the variance of the remnant amounts to some 90 1 of the variance of the steering
signal when using a linear controller model, and is still 35 per cent when a non-linear model is used. So the only possibility at
the present state of art is not to try to model the human control, but to replace the helmsman by an automatic controller. By
optimizing this controller for a given task, a reliable reference can at least be found, which can be completed, when more accurate data is available on the manual steering.
The problem of navigation is also closely connected to human behaviour. Incertain cases the position fixing accuracy and resolution can be fairly easily determined. Examples can be mentioned: leading line with lights, radar návigation, and Oecca navigation. It has been shown, however, that not only the accuracy of information counts, but to alarge extent also the way o-F presenting it to the helmsman or pilot /50/. The most difficult area is certainly the navigation based on purely visual perception, as here only very rough assumptions can be made on the accuracy and resolution. The qualitative aspects of navigation accuracy can,
however, be studied incorporating appropriate dead.zones and stocastic errors in the simulated position fixing system.
3.5.2 The closed-loop control system
CONTROL PARAMETERS
CONTROL
SlATE VECTOR ESTIMATE
CONTROL ACTIONS SENSI NG SYSTEM DISTURBANCES SM P AND WATERWAY
Fig. 3 Block diagram of a steered ship.
a waterway. The ship and waterway are taken as a single block, as the equations of ship motions in restricted waters actually are site dependent as discussed in Section 3.3. The disturbances directly affecting the ship state are those discussed in Section 3.4. The control actions, which are coming from the control system, consist in the simplest case of only the ordered rudder angle, but normally also of engine setting(s), propeller pitch setting(s) and even bow thrusters, tugs etc.. In the present study only tha rudder control is considered, however.
The ship state is sensed by the controlling element i.e. the piloti helmsman combination in most cases. The pilot or mate has the main responsibility as the helmsman usually only has to observe the ship's heading using the compass. The pilot tries to estimate the ship's state by making visual observations directly or using aids like radar, Decca, log, and compass. Many new vessels even have a rate gyro and a Doppler log, which directly indicate the rate of
turning and both velocity components respectively. As the state estimating methods are affected by atmospheric, local, or other disturbances, the result differs more or less from the actual state of the ship. As now the control actions taken by the ship
D I STURBAN CES
controller are based on at least slightly erroneous estimates, it is obvious that this can as well lead to accidents, as wrong ship handling with precise state estimate. The weakest point in
navigation is certainly the position estimation, expecially when the fairway edges are not adequately marked.
3.5.3 Ship positioning
The following of a straight leading line is the most elementary kind of ship manoeuvring. The leading line is normally defined by leading lights, but other Forms, like an optical definition by channel banks or buoyage arrangements are also used. Electronic navigation aids can also provide the navigator with suitable transversal position indication, which can have a considerably better accuxacy than the visual judgement of distance from fairway markings or leading lights. The electronic navigation aids
generally have a randomly distributed error like the skywave in Decca positioning, which can easily be added to the required manoeuvring lane attained by following the indicated line.
In visual navigation the transverse position error has a threshold value, below which no deviation from the desired line can be perceived. This is clearly demonstrated by the charabteristics of the leading line of lights /53/, but it is also evident for example in channel navigation, that a clear sideways transfer is required before it can be detected by the operator. As the maximum accuracy of the visual judgment of distances is of the order of 3 bits /54/, only fairly inaccurate position estimation can be expected by eyes alone. On the other hand, if there is ample width of waterway available, the operator can tolerate rather large deviations from
the desired line and thus create another type of self-set threshold to cause control action. The simplest wy to describe this type of control behaviour is, as already stated, to use a dead-zone
function in the computation of the observed value of the transverse distance of the vessel from the line.
3.6.1 General principles
The Monte-Carlo analysis involves simulating a system a large number of times. Here only the direct simulation is considered, but the method has been succesfully used for a wide variety of problems /55/. In aerospace research the Monte-Carlo simulation has been used since the emergence of computers. It has been used for trajectory scatter and mission success estimations /56/. A steered ship transiting a given waterway or area of congested water is also a typical system where a large number of factors contribute to deviations from a nominal path. Thus the method was applied to study the aspects of waterway dimensioning.
The most important drawback of the Monte-Carlo method is the considerable amount of computer time required. There are, however, some shortcut methods, which are able to considerably reduce the computation time /57, 55/. Such methods still require comparative direct simulations to test their validity and accuracy in the case of the problem at hand. Thus it was decided to apply direct
simulation in the present study in order to exclude possible error sources in the case of merely applying some simplified method.
The input values in the Monte-Carlo simulations are the parameters, which in reality obey some statistical distribution. The determin-ation of those distributions is the first job in the analysis, and it will be discussed more in detail later on. Random number
generating technique is applied to create appropriate statistical distributions for the simulation process. The resulting rnanoeuvring results are stored by the computer. Then they can be analyzed statistically to give the probability distributions and their parameters of the required manneuvring space at the desired point of the fairway under study.
The expected error in the standard deviation of a Monte-Carlo simulation output, which obeys a Gaussian distribution is inversely proportional to the square root of the number of oimulatjons /56/. The actual number of simulations used in the
computations was 500. This gives an expected error of 6.2 per cent with 95% confidence level, which was considered tobe a reason-able compromise between cost and accuracy. The number 500 means on the other hand that the confidence level is .994 that at least 99%
of the population lies below the largest value cf the sample.
3.6.2 Distribution selection and parameter estimation
To obtain an appropriate probability distribution for every random variable concerned, all available and relevant data and knowledge must be surveyed. In principle the amound of necessary information
about a variable can vary considerably /59/, if enough data and theoretical understanding is at hand, the selection of a suitable distribution is easy, but even in the opposite case, useful simulation results can be gained, providing a skillful judgement is made. A sensitivity ahalysis is however necessary to check the choice in such cases.
Wind velocity
There is ample data available on wind velocities along Finnish coastal waters. Using data from Katajaluoto, UtU, and Valassaaret weather stations, some 25000 measurements each, the distribution was found to fit the data well. The Weibull-distribution has the form:
U -1 U n
A-c
A-c ni
f(UA) = ) exp [- ) j u where e = location parameter X = scale parameter n = shape parameterTable i shows the estimated distribution parameters at the three
coastal weather stations from the period 1959-72 /60/.
A
>
Table 1. Parameters of the Weibull distributed wind data at three coastal weather stations.
The parameter estimation algorithm applied was published by Askren /51/. The Kolmogorov-Smirnoff test was used to test the fit. The
95 test value was underrun by an order of magnitude.
Wind direction
Data on wind direction distributions has been reported by Venho /62/. As can be expected the SW-winds are the most frequent. Because the direction data is reported in only eight classes, the histogram distribution is applied in the Monte-Carlo analysis. Table 2 shows the wind direction data for the aforementioned coastal weather stations.
The correlation of wind speed and direction is an important
Table 2. Percentages for wind direction of three coastal weather stations /62/
Katajaluoto Uts Valassaaret
N 7.9 12.6 NE 11.0 7.6 15.1 E 13.8 7.7 5.7 SE 10.1 12.3 7.4 S 9.7 11.9 17.0 SW 24.2 22.9 18.7 W 10.2 8.9 9.8 NW 9.7 13.2 6.7 Calm 3.2 2.9 8.0
location shape scale
Katajaluoto 0.0 2.085 7.258
UtS 0.0 2.012 7.386
question, which can hardly be adequately covered, however. This is mainly due to wind measurement techniques applied at the Finnish coastal weather stations. If the Kuuskajaskari station near Rauma on the West coast is taken as an example, it can be noted that the anemometer is placed on a pole 24 m abov.e sea level, but only 4 m above the tree tops. The measurements from the period 1959-73 /60/ show that the mean wind velocity from sea (SW to NW) corresponds to a Beaufort number of 3.6 but the same value for winds from land
(NE to S6) is only 2.7. Before precise wind velocity distribption measurements to reveal the actual shape of the atmospheric
boundary layer are performed, no conclusions about the true wind speed/direction correlation along Finnish coasts can be drawn.
For the purposes of this study it was decided to regard the wind speed and direction as uncorrelated variables. For the general
treatment it is justified /63/ and when using the method for
specific locations where a complete wind data profile is available, even a correlated model can easily be implemented.
Wave forces
As there is actually no wind-wave correlation data from Finnish coastal waters, a complete correlation with wind velocity is
assumed. The fetch is naturally dependent on the wind direction. When adequate measurements are at hand, a more sophisticated model can be developed, where the wind sea and swell can be treated separately according to wind time histories. In th present study the wind data inputs of successive simulations are regarded as independent, as the duration of a harbour approach or exit usually takes less than an hour and the time between sucessive runs is not correlated with weather.
It was not considered necessary to include the water depth
influence on the wave spectra and wave induced ship motions as the waves seldom constitute more than a moderate disturbance to ship steering. On the other hand in most fairway sites the sheltering effect of shoalings, rocks, and islands is common. Thus the wave induced ship motions are in most cases overestimated and the ignored swell influence can be qualitatively compensated.
Water currents
Ship steering qualities
As the flow conditions are rather moderate, the influence of ship steering behaviour is seldom really significant. The act flow characteristics are very poorly known, however. In measurements, related to pollution studies, sea water movemen values o-F the order .5 rn/s have been the top figures /B4/. On other hand as the flow conditions are site dependent, no gene estimate to illustrate the effect of water currents on the mo can be made. For the purposes of this pilot study a normally distributed steady current with zero mean and a standard devi of 0.25 m/s was stipulated. The direction distribution was as uniform between 0° and 1800. For studies concerning a specifi site, the flow conditions must naturally be considered more accurately according to local measurements.
When a specific waterway is to be designed, there is in most a certain ship type, that constitutes, in its largest allowab size, the 'design ship" /1/. Actually there is a size class o
vessels which is interesting from the viewpoint of waterway dimensioning. Nowadays this class consists, with only a few
exceptions, of tankers and bulk carriers. These types tend to the maximum size allowable and have the most lirnìted power resources, and are accordingly equipted with the least contro
force potential. There certainly are areas, where the main pr is dissimilar, for example the dense ferry traffic between Tu and Stockholm. In such cases the main problem is mainly a roar traffic engineering one and only to a lesser extent caused by hydrodynamic difficulties. However, as 70 per cent of the wor trading tonnage consists of tankers and bulk carriers, accord to 1973 Lloyds's statistics, this class of vessels plays a dominating role in most waterway design.
If a waterway with a maximum allowable ship draught of 15 m i taken as object, it can be concluded that the tankers between 50 000 and 150 000 tdw form an interesting class of vessels fr
it on ual t the ra 1 del at io n sumad c cases le f 1 oblem rku
me
ld ing s ornthe viewpoint of dimensioning. The larger vessels must naturally not be fully laden to comply with the draught limit. Now if one wishes to simulate the wide variety of vessels of the afore mentioned class using an oil harbour, he should have an immense amount of manoeuvring data covering all possible ship types and load conditions. This is not feasible, however, partly due to lack of suffici'ent hydrodynamic data and partly due to incomplete information on actual vessels and their running conditions. So a simplified approach had to be chosen.
For the purpose of this study a basic tanker type with normal proportions was taken as a starting point. The size was scaled according to draught requirement. The steering qualities were concentrated to a single variable, the P-factor proposed by Norrbin /65/, which describes the initial rudder responce of the vessel. The P-factor is defined as the ratio of course angle and rudder angle after one sailed ship length from the point where the rudder was set to a 10 degree offset. Mean value of both turning
directions is taken to allow for asymmetry.
Norrbin /65/ has studied the P-factors of different vessels and even proposed the value P = 0.3 as a lower limit for adequate
steerability of a ship. He has collected data from 45 Swedish merchant vessels /66/ and found the distribution of P to be close to normal, but judged the standard deviation to be probably over-estimated. This is understandable as the measurements were carried out by shipmasters and not by professionals. Norrbin also states that the P-value appears to be independent of ship type.
The justification of using the Norrbins P-factor as an only parameter to describe the steering qualities of a certain class of vessels is at the present state of art somewhat vague, but it appears to be the only tool available. The available data limits
its use only to the laden condition. The applied distribution for P has a mean of 0.37 and the standard deviation has been reduced from the original value /66/ of .09 to .07 in order to achieve more realistic extreme values in light of the data in Ref. /65/.
appropriate adjustment of the rudder coefficients in the equations of motion (3.1 to 3.3). A second degree polynomial for the rudder area ratio in terms of P was found to give an adequate accuracy. The linear control-fixed coefficients were accordingly modified following the lines presented by Eda /37/.
Starting point of simulation
In simulator studies a run usually starts from a certain point
(x0 = 0, y = 0). If a run along a straight leading line is taken
as an example, one can see that a mean line is developed, around which the individual runs are normally distributed /14/. The mean
line is not straight, but is of an oscillating character with a period of an order of magnitude ten times ship length /14/ and a certain asymmetry and starting phase due to hydrodynamic asymmetry
caused by propeller rotation. The standard deviation of the paths of the ship center of gravities starts from zero and grows to a certain equilibrium value /14/.
For a simulation of a certain fairway leg the distribution of ship paths is obviously randomly distributed at the starting point. To save computing time it is better to estimate a normally distributed initial deviation and heading to attain the equilibrium state as soon as possible. By experimentation suitable values could be found. As most single screw ships have right-handed screws, the
hydrodynamic asymmetry is to the same direction. On the other hand if the fairway has traffic to both directions the resulting width required regains the symmetry from this cause at least. This can be allowed for also in one way simulation by assuming that a passing vessel has a right- or left-handed screw with 50 % probability. This makes it also unnecessary to simulate fairway
bends in both directions because of ship asymmetry.
3.5.3 Ship control philosophy
As stated earlier in Section 3.5, it is not feasible to accurately model a human ship controller. Thus the only possibility available
is to use a simple controller in the simulation and to try to test the justification of the assumption by comparing the overall performances of manually and automatically controlled vessels. This means for example comparing the necessary fairway widths in identical conditions utilizing bath types of controllers. The simple controller used has the following configuration:
+ T = K ( - p )+K K y + (3.26)
sg p o e d o py o e
The computer program dtermines the deviation of the ship center of gravity from the leading line composed of straight legs and arcs of circles. The heading error and error rate ars also computed. Because the real human controller has only limited accuracy in determining the ectual state, this must be taken in to account in the model, too.
On straight course the error in the reference heading given by the gyrocompass is neglected as it seldom exceeds 20. The transverse distance from the leading line has a certain dead zone caused by the perception ability of the human eye when observing the leading lights, see Section 3.5. In radar navigation tIe accuracy is
correspondingly limited. On the other hand, if there is ample width of waterway awailable, the navigator can set up limiting lines of his own, beyond which he starts to correct the vessel's motion towards the leading line /67/. Thus the most obvious simple model for the observed deviation acting as controller input is a dead space function with a variable width.
In curves a different system is employed. It is assumed that the curve radius can be estimated with good accuracy from a chart or radar screen. In the simulation program the radius of the curve is determined by the ship's position when entering the turn, and the center of the circle coincides with thet of the basic arc. This curve is the base for deviation error signal. The heading error during a turn can only be determined with limited accuracy. The error has been estimated to be proportional to the total change of heading in the curve. This can be regarded as connected to the error in determining the starting point for turning as seen in
s
Fig. 4 Assumed distribution o-f course errors in curve.
An important point is the steering philosophy employed in terminating the turn before coming to the new course. As the controller used can not anticipate the end of the curve, an
overshoot of turning is the result. This is regarded acceptable, however, as the result is merely an esthetic error. An anticipating controller model would also have spoiled the simple controller configuration taken as tentative replacement for the human one.
The control parameters can not be randomly varied, as an unstable behaviour could result. Thus the controller was optimized using the following merit number (see Fig. 2):
- V *2
merit - port + (3.27)
As the rudder area ratio was changed according to the P-factor distribution, the optimal control parameters also change. The weighting with the inverse of the rudder area ratio gives adequate accuracy in most cases and only very small rudder areas require a progressive dependence of yaw rate control gain from the inverse of the rudder area ratio.
3.6.4 Random number generation
As the objective of this study is well characterized by the term, comparative simulation /55/, the random number generation must be able to generate repeatedly, exactly the same sequence o-F
pseudorandom numbers. This assures the best accuracy when comparing the results of two slightly different situations, as variations caused by sampling are minimized. A computer independent RN generator is certainly the ideal choice as it supplies the same sequence even on different installations. Many rendom number generators in statistical machine dependent program packages use a time dependent starting number for the program, which prevents the repetition of the same sequence in subsequent runs. Thus they are completely unsuitable for purposes of this kind of Monte-Carlo simulation. A suitable machine independent pseudorandom number generation algorithm was published by Mc Grath /88/. The same paper includes a completd library of algorithms for the generation of random numbers with prescribed probability distribution as required for the simulation.
3.7 Computational scheme
3.7.1 Choosing the simulation system
During the early stages of the present study considerable effort was directed towards application of special continuous system modelling languages CSMF and MIMIC /69, 70/. As only the latter was available in the UNIVAC 1108 installation used, all experience gained is from MIMIC.
The programming oF the actual equations etc. in MIMIC is very easy and straightforward. The block-oriented form of the language especially, makes the programming of integrations very convenient. The main program can be completed with five FORTRAN-subprograms, which allow desired special functions and features to be included.
The drawbacks are rather severe, however. The block-oriented quasi-parallel operation of a MIMIC-program spoils the clear
to exchange data between the program units, as only seven
parameters are allowed in a subprogram call, and no common area is available in the main program. Perhaps the most serious weakness of the MIMIC-language was the long computing time, which was by more than an order of magnitude more than later realized by
FORTRAN-programming. The reason was the small minimum time step required to attain stability in integrations.
Due to the drawbacks of the MIMIC, it was abandoned, and it was decided to perform all the simulation using simple FORTRAN-programs. By a suitable subdivision of the task into subprograms the structure can be held clear and handy, simultaneously a practical division into functional blocks was realized. The
common area makes the transfer of data very convenient. The clear sequential nature of a FORTRAN program also makes the timing o-F events easy.
3.7.2 Simulation details
The functional lay-out of the simulation program can be seen in Fig. 5. The direct information flow sequence during an integration cycle is indicated by numbered arrows. Simultaneously all necessary information is continuously distributed via the common variables. The integrations are performed by the UNIVAC-library routine PKDE, which uses a modified fourth order Runge-Kutta method. It requires only two additional function evaluations per time step when
compared to the simple Euler-method as applied by Strm-Tejsen /22/. Thus the same computing time is attained with the R-K method as with the Euler method using a third of the time step. When the time step for the R-K method is chosen to be about 10 of the time constant of the ship, the computing error is by more than an oçder of magnitude smaller with the same computing time /71/. The self-starting ability of the R-K method is also important for a general purpose simulator.
The main program and the subprogram, as well as all other programs used for purposes of this study are listed in Appendix III. The essence o-F most subprograms of' the simulation scheme is written
N Ti al RKDE integration SHIP derivatives TRACK MAI N program ENWironment OBServoti on PILOT LANE
